Potentially wrong q-expansions

[Arnau Padrés Masdemont]

Hi all,

I believe the following q-expansions could be wrong. 

- In https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/229/1/d/b/, the coefficient of q^8 should be 0. 

- In https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/633/1/m/b/, the coefficient of q^8 should be -z^9 + z^7 - z^5 + z^3 instead of -z^9 + z^7 - z^5 - z^3

The reason why I believe they might be wrong is that the data in SageMath suggests so. For instance, in the first case I use Sage to take a basis of the space of cusp forms with the character of 229.1.d.b, which has two elements. All the coefficients of 229.1.d.b coincide with a linear combination of the basis elements (as they should) except from q^8. The same happens with 663.1.m.b. However, I do not know if the mistake comes from Sage or from LMFDB. 

I hope this helps,

Arnau

[David W. Farmer]

For the first one, we can see by eye that something is wrong,
because the q^8 coefficient of the trace form does not equal
the sum of the coefficients of q^8 of two newforms in the space.

Also, since 2 is good, the q^2 coefficient determines all the
q^4, q^8, q^16, etc, coefficients.

I suspect that this is just a display error, because if you
look a the two associated L-functions, those appear to have
correct coefficients.

Regards,

David

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[Andrew Sutherland]

Thank you for reporting this, I agree the q-expansion that is being displayed is incorrect in both cases.

The underlying data stored in the database appears to be mathematically correct but contains some redundancy that is not being handled correctly by the display code: the entry for q^8 for the first form is stored as 1*z^0 -1*z^0, where z=zeta_4, see https://www.lmfdb.org/api/mf_hecke_nf/21892) and the entry for q^8 for the second form is stored as 2z^3 - z^3 - z^5 + z^7 -z^9 , where z =- zeta_20, see https://www.lmfdb.org/api/mf_hecke_nf/45775.

I've created an issue for this at https://github.com/LMFDB/lmfdb/issues/6074.  It should not be hard to fix the display issue.

Andrew Sutherland